Connectivity of discrete planes

• Authors: Valentin E. Brimkov;Reneta P. Barneva
• Venue: Theoretical Computer Science - Combinatorics of the discrete plane and tilings
• Year: 2004

Studying connectivity of discrete objects is a major issue in discrete geometry and topology. In the present work, we deal with connectivity of discrete planes in the framework of Réveillès analytical definition (Thèse d'État, Université Louis Pasteur, Strasbourg, France, 1991). Accordingly, a discrete plane is a set P(a, b, c, µ, ω) of integer points (x, y, z) satisfying the Diophantine inequalities 0 ≤ ax + by + cz + µ Z estimates the plane intercept while ω ∈ N is the plane thickness. Given three integers (plane coefficients) a, b, and c with 0 ≤ a ≤ b ≤ c, one can seek the value of ω beyond which the discrete plane P(a, b, c, µ, ω) is always connected. We call this remarkable topological invariable the connectivity number of P(a, b, c, µ, ω) and denote it Ω(a, b, c). Despite several attempts over the last 10 years to determine the connectivity number, this is still an open question. In the present paper, we propose a solution to the problem. For this, we first investigate some combinatorial properties of discrete planes. These structural results facilitate the deeper understanding of the discrete plane structure. On this basis, we obtain a series of results, in particular, we provide an explicit solution to the problem under certain conditions. We also obtain exact upper and lower bounds on Ω(a, b, c) and design an O(alogb) algorithm for its computation.